Prove walk splitting

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice/Builder.agda

@@ -2,7 +2,7 @@ module Lattice.Builder where
 
 open import Lattice
 open import Equivalence
-open import Utils using (Unique; push; empty; Unique-append; All¬-¬Any; ¬Any-map; fins; fins-complete)
+open import Utils using (Unique; push; empty; Unique-append; Unique-++⁻ˡ; Unique-++⁻ʳ; All¬-¬Any; ¬Any-map; fins; fins-complete)
 open import Data.Nat as Nat using (ℕ)
 open import Data.Fin as Fin using (Fin; suc; zero; _≟_)
 open import Data.Maybe as Maybe using (Maybe; just; nothing; _>>=_; maybe)
@@ -12,12 +12,12 @@ open import Data.List.NonEmpty using (List⁺; tail; toList) renaming (_∷_ to
 open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_; mapWith∈ to mapWith∈ˡ)
 open import Data.List.Membership.Propositional.Properties using () renaming (∈-++⁺ʳ to ∈ˡ-++⁺ʳ; ∈-++⁺ˡ to ∈ˡ-++⁺ˡ; ∈-cartesianProductWith⁺ to ∈ˡ-cartesianProductWith⁺)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
-open import Data.List.Relation.Unary.All using (All; []; _∷_; map; lookup)
-open import Data.List.Relation.Unary.All.Properties using () renaming (++⁺ to ++ˡ⁺)
+open import Data.List.Relation.Unary.All using (All; []; _∷_; map; lookup; zipWith)
+open import Data.List.Relation.Unary.All.Properties using () renaming (++⁺ to ++ˡ⁺; ++⁻ʳ to ++ˡ⁻ʳ)
 open import Data.List using (List; _∷_; []; cartesianProduct; cartesianProductWith; foldr) renaming (_++_ to _++ˡ_)
-open import Data.List.Properties using () renaming (++-conicalʳ to ++ˡ-conicalʳ)
+open import Data.List.Properties using () renaming (++-conicalʳ to ++ˡ-conicalʳ; ++-identityʳ to ++ˡ-identityʳ; ++-assoc to ++ˡ-assoc)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
-open import Data.Product using (Σ; _,_; _×_; proj₁; proj₂)
+open import Data.Product using (Σ; _,_; _×_; proj₁; proj₂; uncurry)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
@@ -45,13 +45,25 @@ record Graph : Set where
         done : ∀ {n : Node} → Path n n
         step : ∀ {n₁ n₂ n₃ : Node} →  (n₁ , n₂) ∈ˡ edges → Path n₂ n₃ → Path n₁ n₃
 
+    data IsDone : ∀ {n₁ n₂} → Path n₁ n₂ → Set where
+        isDone : ∀ {n : Node} → IsDone (done {n})
+
     _++_ : ∀ {n₁ n₂ n₃} → Path n₁ n₂ → Path n₂ n₃ → Path n₁ n₃
     done ++ p = p
     (step e p₁) ++ p₂ = step e (p₁ ++ p₂)
 
-    ++done : ∀ {n₁ n₂} (p : Path n₁ n₂) → p ++ done ≡ p
-    ++done done = refl
-    ++done (step e∈edges p) rewrite ++done p = refl
+    ++-done : ∀ {n₁ n₂} (p : Path n₁ n₂) → p ++ done ≡ p
+    ++-done done = refl
+    ++-done (step e∈edges p) rewrite ++-done p = refl
+
+    ++-assoc : ∀ {n₁ n₂ n₃ n₄} (p₁ : Path n₁ n₂) (p₂ : Path n₂ n₃) (p₃ : Path n₃ n₄) →
+                 (p₁ ++ p₂) ++ p₃ ≡ p₁ ++ (p₂ ++ p₃)
+    ++-assoc done p₂ p₃ = refl
+    ++-assoc (step n₁,n∈edges p₁) p₂ p₃ rewrite ++-assoc p₁ p₂ p₃ = refl
+
+    IsDone-++ˡ : ∀ {n₁ n₂ n₃} (p₁ : Path n₁ n₂) (p₂ : Path n₂ n₃) →
+                 ¬ IsDone p₁ → ¬ IsDone (p₁ ++ p₂)
+    IsDone-++ˡ done _ done≢done = ⊥-elim (done≢done isDone)
 
     interior : ∀ {n₁ n₂} → Path n₁ n₂ → List Node
     interior done = []
@@ -70,8 +82,17 @@ record Graph : Set where
     ...   | step _ p | inj₁ IH rewrite IH = inj₁ refl
     ...   | step _ p | inj₂ IH rewrite IH = inj₂ refl
 
+    interior-++ : ∀ {n₁ n₂ n₃} → (p₁ : Path n₁ n₂) →  (p₂ : Path n₂ n₃) →
+                    ¬ IsDone p₁ → ¬ IsDone p₂ →
+                    interior (p₁ ++ p₂) ≡ interior p₁ ++ˡ (n₂ ∷ interior p₂)
+    interior-++ done _ done≢done _ = ⊥-elim (done≢done isDone)
+    interior-++ _ done _ done≢done = ⊥-elim (done≢done isDone)
+    interior-++ (step _ done) (step _ _) _ _ = refl
+    interior-++ (step n₁,n∈edges p@(step n,n'∈edges p')) p₂ _ p₂≢done
+        rewrite interior-++ p p₂ (λ {()}) p₂≢done = refl
+
     SimpleWalkVia : List Node → Node → Node → Set
-    SimpleWalkVia ns n₁ n₂ = Σ (Path n₁ n₂) (λ p → Unique (interior p) × All (λ n → n ∈ˡ ns) (interior p))
+    SimpleWalkVia ns n₁ n₂ = Σ (Path n₁ n₂) (λ p → Unique (interior p) × All (_∈ˡ ns) (interior p))
 
     SimpleWalk-extend : ∀ {n₁ n₂ n₃ ns} →  (w : SimpleWalkVia ns n₁ n₂) → (n₂ , n₃) ∈ˡ edges →  All (λ nʷ →  ¬ nʷ ≡ n₂) (interior (proj₁ w)) → n₂ ∈ˡ ns → SimpleWalkVia ns n₁ n₃
     SimpleWalk-extend (p , (Unique-intp , intp⊆ns)) n₂,n₃∈edges w≢n₂ n₂∈ns
@@ -81,7 +102,54 @@ record Graph : Set where
             with intp++[n₂]⊆ns ←  ++ˡ⁺ intp⊆ns (n₂∈ns ∷ [])
             rewrite sym intp'≡intp++[n₂] = (p' , (subst Unique (sym intp'≡intp++[n₂]) (Unique-append (¬Any-map sym (All¬-¬Any w≢n₂)) Unique-intp) , intp++[n₂]⊆ns))
 
-    postulate SplitSimpleWalkVia : ∀ {n n₁ n₂ ns} (w : SimpleWalkVia (n ∷ ns) n₁ n₂) →  (Σ (SimpleWalkVia ns n₁ n × SimpleWalkVia ns n n₂) λ (w₁ , w₂) → proj₁ w₁ ++ proj₁ w₂ ≡ proj₁ w) ⊎ (Σ (SimpleWalkVia ns n₁ n₂) λ w' → proj₁ w' ≡ proj₁ w)
+    ∈ˡ-narrow : ∀ {x y : Node} {ys : List Node} → x ∈ˡ (y ∷ ys) → ¬ y ≡ x → x ∈ˡ ys
+    ∈ˡ-narrow (here refl) x≢y = ⊥-elim (x≢y refl)
+    ∈ˡ-narrow (there x∈ys) _ = x∈ys
+
+    SplitSimpleWalkViaHelp : ∀ {n n₁ n₂ ns} (nⁱ : Node)
+                                            (w : SimpleWalkVia (n ∷ ns) n₁ n₂)
+                                            (p₁ : Path n₁ nⁱ) (p₂ : Path nⁱ n₂) →
+                                            ¬ IsDone p₁ → ¬ IsDone p₂ →
+                                            All (_∈ˡ ns) (interior p₁) →
+                                            proj₁ w ≡ p₁ ++ p₂ →
+                                            (Σ (SimpleWalkVia ns n₁ n × SimpleWalkVia ns n n₂) λ (w₁ , w₂) → proj₁ w₁ ++ proj₁ w₂ ≡ proj₁ w) ⊎ (Σ (SimpleWalkVia ns n₁ n₂) λ w' → proj₁ w' ≡ proj₁ w)
+    SplitSimpleWalkViaHelp nⁱ w done _ done≢done _ _ _ = ⊥-elim (done≢done isDone)
+    SplitSimpleWalkViaHelp nⁱ w p₁ done _ done≢done _ _ = ⊥-elim (done≢done isDone)
+    SplitSimpleWalkViaHelp {n} {ns = ns} nⁱ w@(p , (Unique-intp , intp⊆ns)) p₁@(step _ _) p₂@(step {n₂ = nⁱ'} nⁱ,nⁱ',∈edges p₂') p₁≢done p₂≢done intp₁⊆ns p≡p₁++p₂
+        with intp≡intp₁++[n]++intp₂ ← trans (cong interior p≡p₁++p₂) (interior-++ p₁ p₂ p₁≢done p₂≢done)
+        with nⁱ∈n∷ns ∷ intp₂⊆n∷ns ← ++ˡ⁻ʳ (interior p₁) (subst (All (_∈ˡ (n ∷ ns))) intp≡intp₁++[n]++intp₂ intp⊆ns)
+        with nⁱ ≟ n
+    ...   | yes refl
+            with Unique-intp₁ ← Unique-++⁻ˡ (interior p₁) (subst Unique intp≡intp₁++[n]++intp₂ Unique-intp)
+            with (push n≢intp₂ Unique-intp₂) ← Unique-++⁻ʳ (interior p₁) (subst Unique intp≡intp₁++[n]++intp₂ Unique-intp)
+            = inj₁ (((p₁ , (Unique-intp₁ , intp₁⊆ns)) , (p₂ , (Unique-intp₂ , zipWith (uncurry ∈ˡ-narrow) (intp₂⊆n∷ns , n≢intp₂)))) , sym p≡p₁++p₂)
+    ...   | no nⁱ≢n
+            with p₂'
+    ...       | done
+                = let
+                      -- note: copied with below branch. can't use with <- to
+                      --       share and re-use because the termination checker loses the thread.
+                      p₁' = (p₁ ++ (step nⁱ,nⁱ',∈edges done))
+                      n≢nⁱ n≡nⁱ = nⁱ≢n (sym n≡nⁱ)
+                      intp₁'=intp₁++[nⁱ] = subst (λ xs → interior p₁' ≡ interior p₁ ++ˡ xs) (++ˡ-identityʳ (nⁱ ∷ [])) (interior-++ p₁ (step nⁱ,nⁱ',∈edges done) p₁≢done (λ {()}))
+                      intp₁++[nⁱ]⊆ns = ++ˡ⁺ intp₁⊆ns (∈ˡ-narrow nⁱ∈n∷ns n≢nⁱ ∷ [])
+                      intp₁'⊆ns = subst (All (_∈ˡ ns)) (sym intp₁'=intp₁++[nⁱ]) intp₁++[nⁱ]⊆ns
+                      -- end shared with below branch.
+                      Unique-intp₁++[nⁱ] = Unique-++⁻ˡ (interior p₁ ++ˡ (nⁱ ∷ [])) (subst Unique (trans intp≡intp₁++[n]++intp₂ (sym (++ˡ-assoc (interior p₁) (nⁱ ∷ []) []))) Unique-intp)
+                  in inj₂ ((p₁ ++ (step nⁱ,nⁱ',∈edges done) , (subst Unique (sym intp₁'=intp₁++[nⁱ]) Unique-intp₁++[nⁱ] , intp₁'⊆ns)) , sym p≡p₁++p₂)
+    ...       | p₂'@(step _ _)
+                = let p₁' = (p₁ ++ (step nⁱ,nⁱ',∈edges done))
+                      n≢nⁱ n≡nⁱ = nⁱ≢n (sym n≡nⁱ)
+                      intp₁'=intp₁++[nⁱ] = subst (λ xs → interior p₁' ≡ interior p₁ ++ˡ xs) (++ˡ-identityʳ (nⁱ ∷ [])) (interior-++ p₁ (step nⁱ,nⁱ',∈edges done) p₁≢done (λ {()}))
+                      intp₁++[nⁱ]⊆ns = ++ˡ⁺ intp₁⊆ns (∈ˡ-narrow nⁱ∈n∷ns n≢nⁱ ∷ [])
+                      intp₁'⊆ns = subst (All (_∈ˡ ns)) (sym intp₁'=intp₁++[nⁱ]) intp₁++[nⁱ]⊆ns
+                      p≡p₁'++p₂' = trans p≡p₁++p₂ (sym (++-assoc p₁ (step nⁱ,nⁱ',∈edges done) p₂'))
+                  in SplitSimpleWalkViaHelp nⁱ' w p₁' p₂' (IsDone-++ˡ _ _ p₁≢done) (λ {()}) intp₁'⊆ns p≡p₁'++p₂'
+
+    SplitSimpleWalkVia : ∀ {n n₁ n₂ ns} (w : SimpleWalkVia (n ∷ ns) n₁ n₂) →  (Σ (SimpleWalkVia ns n₁ n × SimpleWalkVia ns n n₂) λ (w₁ , w₂) → proj₁ w₁ ++ proj₁ w₂ ≡ proj₁ w) ⊎ (Σ (SimpleWalkVia ns n₁ n₂) λ w' → proj₁ w' ≡ proj₁ w)
+    SplitSimpleWalkVia (done , (_ , _)) = inj₂ ((done , (empty , [])) , refl)
+    SplitSimpleWalkVia (step n₁,n₂∈edges done , (_ , _)) = inj₂ ((step n₁,n₂∈edges done , (empty , [])) , refl)
+    SplitSimpleWalkVia w@(step {n₂ = nⁱ} n₁,nⁱ∈edges p@(step  _ _) , (push nⁱ≢intp Unique-intp , nⁱ∈ns ∷ intp⊆ns)) = SplitSimpleWalkViaHelp nⁱ w (step n₁,nⁱ∈edges done) p (λ {()}) (λ {()}) [] refl
 
     postulate findCycle : ∀ {n₁ n₂} (p : Path n₁ n₂) →  (Σ (SimpleWalkVia (proj₁ nodes) n₁ n₂) λ w → proj₁ w ≡ p) ⊎ (Σ Node (λ n → Σ (SimpleWalkVia (proj₁ nodes) n n) λ w →  ¬ proj₁ w ≡ done))